How to animate object in local space?
Yes, but a much simpler (conventional) way is to have a setup like this.

Empty2 -&gt; Empty1 -&gt; Object, where the arrow represents a 'parent of' relationship.

The Empty2 is a parent of Empty1, and Empty 1 is a parent of Object. Then when you keyframe Empty1, Object will move accordingly. On top of that you can then keyframe Empty2 and Empty1 (and therefor Object) will move in relative orientations.